🤬 To foul or not to foul

In FIBA 3x3, fouling is a tactic used to prevent the offensive team from scoring easy baskets when the defending team has less than 7 fouls. Disregarding unsportsmanlike and technical fouls for the moment and assuming the fouled team is the team on offense, for fouls 1 to 6, a checkball and a shot clock reset are given to the fouled team but no free throw is given unless the foul occured on a shot attempt. For fouls 7 to 9, two free throws are awarded to the fouled team. For fouls 10 and above, two free throws plus possession of the ball are given to the fouled team.

Since there is a sharp tradeoff for fouling before/after foul 6, some teams have developed fouling strategies to take advantage of this discrepancy. From the perspective of the player guarding a mismatch inside the restricted area, would it make more sense to foul the opponent if one's team has less than 6 fouls or give up one point in order to set up the transition offense? This also begs the question of how many points are each foul worth?

One way to quantify this is to divide the number of points from made free throws over total number of fouls. From the 2021 and 2022 Pro Circuit (these are the World Tour and Challenger events) and Women's Series, each foul is worth slightly more than 0.3 points and a foul is committed on 17-20% of all possessions.

Factoring into the fact that fouls 1 to 6 results in a checkball for the offensive team and a checkball possession is worth around 0.54 points (courtsey of New Zealand 3x3's Piet Van Hasselt @pvhbball), this results in roughly 0.85-0.9 points overall per foul.

If the foul occured on a shot attempt, one free throw is awarded for shots inside the arc and two are awarded for shots outside the arc. Using an estimate of 0.65 points per free throw from the overall free throw shooting percentage for a season, this translates to roughly 0.95 points per foul on shot attempts inside the arc and 1.65 points per foul on shot attempts outside the arc. Needless to say, it doesn't make sense to foul when the opponent is shooting regardless of the foul situation but it might be worthwhile to foul before foul number 6 to avoid giving away easy one pointers.

Comparison of teams from 2021 and 2022 Pro Circuit and Women's Series

Of all the teams that played at least 10 games in the 2021 or 2022 seasons, there is a positive correlation between free throw made per foul and free throw percentage as expected. It is also interesting to see a positive correlation between free throw made per foul and free throw attempts per possession and the author does not have a good explanation at this time. The horizontal and vertical dashed lines in the plots below represent the overall value for each team statistic.